Abstract
We consider the existence problem for ‘Steiner networks’ (trivalent graphs with 2π/3 angles at each junction) in strictly convex domains, with ‘Neumann’ boundary conditions. For each of the three possible combinatorial possibilities, sufficient conditions on the domain are derived for existence. In addition, in each case explicit examples of nonexistence are given.
Similar content being viewed by others
References
Bronsard L., Reitich F.: On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rational Mech. Anal. 124(4), 355–379 (1993)
Ikota R., Yanagida E.: Stability of stationary interfaces of binary-tree type. Calc. Var. Partial Differ. Equ. 22(4), 375–389 (2005)
Ivanov, A., Tuzhilin, A.: Branching Solutions to One-Dimensional Variational Problems. World Scientific Publishing Co., River Edge, 2001. xxii+342 pp. ISBN:981-02-4060-0
Mantegazza C., Novaga M., Tortorelli V. (2004) Motion by curvature of planar networks. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(3), no. 2, 235–324
Mese C., Yamada S.: The parameterized Steiner problem and the singular plateau Problem via energy. Trans. Am. Math. Soc. 358(7), 2875–2895 (2006)
Tabachnikov S.: The four-vertex theorem revisited. Am. Math. Mon. 102(10), 912–916 (1995)
Yaglom I.M., Boltianski V.G.: Convex Figures. Rinehart and Winston, Holt (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Müller
Rights and permissions
About this article
Cite this article
Freire, A. The Existence Problem for Steiner Networks in Strictly Convex Domains. Arch Rational Mech Anal 200, 361–404 (2011). https://doi.org/10.1007/s00205-011-0414-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0414-2